We note that from Eq. (REF ) the eigenvalue-magnitudes are symmetric due to the cosine function, implying that \(\lambda _n = \lambda _{N-n+1}\;\forall \,n>0\)  and \(\lambda _0 = 0\)  for any cyclic graph. This implies that almost every eigenvalue is (at least) doubly degenerate, except for \(\lambda _0 = 0\) . Also, we note that the smallest non-zero eigenvalue, \(\lambda _F\)  (known as Fiedler eigenvalue [1]} or algebraic connectivity), or the maximum eigenvalue, \(\lambda _M\) , of a given cyclic graph, can be different than \(\lambda _1\)  or \(\lambda _{N/2}\)  from Eq. (REF ), respectively.
